Calista Bernard - Applications of twisted homology operations for E_n-algebras

An E_n-algebra is a space equipped with a multiplication that is commutative up to homotopy. Such spaces arise naturally in geometric topology, number theory, and mathematical physics; some examples include classifying spaces of braid groups, spaces of long knots, and classifying spaces of general linear groups. The structure of an E_n-algebra gives rise to operations on the homology of a space, and these operations prove quite useful for studying homology. In the 70s F. Cohen and J.P. May gave a complete description of operations on the mod p homology of E_n-algebras, and more recently I have worked on generalizing their results to homology with twisted coefficients. In this talk I will give a brief introduction to E_n-algebras and the theory of operations, and I will then discuss work in progress on applications of this theory to studying the homology of special linear groups SL_n(Z) and to studying the twisted homology of mixed braid groups.